The tables below show the records obtained with Glop for the games of Sprouts, Cram and Dots-and-Boxes.

We indicate in the tables the name of the author(s) if the result was obtained by another team first. The date indicates when the outcome was first computed by Glop or by another team. Please contact us if you find an error in the names or dates for the results of another team.

We also provide the smallest known databases, ie the databases with the minimal known number of losing couples (position, nimber) that allows you to check with Glop the outcome (win or loss) of the starting position. The number of positions indicated below all correspond to the smallest known Glop database at the current time, which is usually much smaller than when the result was first computed. In fact, databases are frequently reduced when we improve Glop, which explains why we continue to find smaller databases many years after the first computation.

The number of positions is not the actual number of positions we needed for our computation, because we have also computed all the positions in the game tree of the 6-spot game. See our article for more details.

The files can be used in the “Rct Misere” tab of Glop 2.0 or higher. “n6-pos-rct” must be loaded in the “Pos/Rct” database, “n6-rct-ch” in the “Rct/Children” database, and any other file in the “Rct Misere” database.

We detail here the records for the game of Sprouts in normal version, played on an arbitrary compact surface. There are two very different kinds of compact surfaces other than the plane (which is equivalent to the sphere) :

orientable ones: a torus with an arbitrary number of holes.

unorientable ones: an arbitrary number of projective planes “glued” together.

In the table below, we only indicate the nimbers of the positions for a given number of starting spots and a given surface (reminder : the position is losing if and only if the nimber is 0).

As far as we know, Glop is the only program able to compute the outcome of Sprouts position on an arbitrary compact surface. We don’t indicate the dates of computation. Our first results date back from November 2008.

S = Sphere (equivalent to the plane).

Tn = Torus with n holes.

Pn = n Projective planes glued together (note that P2 is the famous Klein Bottle).

The following table indicates the results obtained for surfaces with an orientable or non-orientable genus of less than 8 and for positions up to 14 initial spots.

spots

P8

P7

P6

P5

P4

P3

P2

P1

S

T1

T2

T3

T4

T5

T6

T7

T8

T9

2

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

3

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

4

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

5

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

6

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

8

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

9

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

10

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

0

>1

0

>1

0

>3

0

1

1

1

1

1

1

1

1

1

1

1

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

13

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

14

1

>2

1

>2

1

>2

0

0

0

0

0

0

0

0

0

0

0

0

From the right part of the table (sphere and torus with n holes), we can conjecture that for a given number of spots, the nimber is the same on all orientable surfaces.

Ok. Here we are. Glop has computed that the outcome of the 9-spot Sprouts game in misère version on a Klein Bottle is a win. We are still wondering whether it is an essential result for mankind, but we put here the table of misère Sprouts outcome on arbitrary compact surfaces, just in case.

The files can be used in the “Rct Misere” tab of Glop 2.0 or higher. “n6-pos-rct” must be loaded in the “Pos/Rct” database, “n6-rct-ch” in the “Rct/Children” database, and any other file in the “Rct Misere” database.

We detail here the records for the game of Cram, played in the normal version. There exists a simple symmetry strategy for board of evenxeven dimensions, which implies that these boards are losing, and of nimber 0. With a similar symmetry strategy, we can deduce that boards of evenxodd dimensions are winning. However, it implies nothing about the exact value nimber, which can be any number greater than 1, so computing the value of the nimber is of interest.

In the following tables, we have listed all the known results that are not directly implied by the strategy symmetry.

The first table indicates the results known about 3xn boards, and the next table what we know about larger boards. For some boards, we have only been able to compute the outcome (win/loss), or a lesser bound on the nimber value.

We detail here the records for the game of Cram, played in the misère version. In the case of the misère version, the symmetry strategies don’t apply. Computation is then interesting for all sizes of boards.

In the following, we give the misère grundy-value, defined for a position P as the unique n such that P + n is a loss in the misère version. However, it should be noted that its meaning is not the same as the nimber of the normal version. In particular, the misère Grundy-value does not characterize a misère position completely, and contrary to the nimbers of the normal version, the nimber-addition of two misère Grundy-values is not possible.

The first table indicates the known results about 3xn boards. Quite unexpectedly, the 3xn misère version of Cram behaves more regularly than the normal version. From the computed values, we can conjecture that the sequence of misère Grundy-value for 3xn misère Cram is periodic, with a period of length 3.

The reduced canonical tree (RCT) of the 3×8 board have been computed first. We indicate the number of extra positions needed to reach a given result once the RCT of the 3×8 board has been computed. If we don’t indicate the number of positions, it means that the result is immediate once the RCT of the 3×8 board is known.

The files can be used in the “Rct Misere” tab of Glop 2.1. Since Cram symmetries have been optimized in version 2.1, these files cannot be used with version 2.0. “cram3x8-PosRct” must be loaded in the “Pos/Rct” database, “cram3x8-RctChildren” in the “Rct/Children” database, and any other file in the “Rct Misere” database.

Note : The current files are not compatible with the latest 2.2 version of Glop. New files for version 2.2 coming soon.

The following table indicates what we know about larger misère Cram boards. The reduced canonical tree (RCT) of the 5×5 board have been computed first. We indicate the number of extra positions needed to reach a given result once the RCT of the 5×5 board has been computed. If we don’t indicate the number of positions, it means that the result is immediate once the RCT of the 5×5 board is known.

The files can be used in the “Rct Misere” tab of Glop 2.1. Since Cram symmetries have been optimized in version 2.1, these files cannot be used with version 2.0. “cram5x5-PosRct” must be loaded in the “Pos/Rct” database, “cram5x5-RctChildren” in the “Rct/Children” database, and any other file in the “Rct Misere” database.

Note : The current files are not compatible with the latest 2.2 version of Glop. New files for version 2.2 coming soon.

We detail here the records for the game of Dots-and-Boxes. The best results known before our work seems to be from David Wilson. Please note that there are some results not mentionned on Wilson’s web site, but which could probably be computed with his program. In this case, we have chosen to leave the result with no author (-). We consider a result from Glop as new only for positions with strictly more than 40 edges in american version, and at least 40 edges in icelandic or swedish version.

The board dimensions in the tables are given as a number of boxes. Please be careful because the size of Dots-and-Boxes positions is defined in terms of boxes or in terms of dots, depending on the people. A board of NxM boxes is the same as a board of (N+1)x(M+1) dots.

The score of a position is given as a couple s1/s2 where s1 is the maximum number of boxes that the first player is sure to capture if he plays perfectly, and s2 is the maximum number of boxes that the second player is sure to capture if he plays perfectly too. s1+s2 is always equals to the total number of available boxes, that is NxM on a NxM board.

Note that if s1>s2, the outcome is a win (the first player can win). If s1<s2. the outcome is a loss (the first player cannot win). And if s1=s2, the outcome depends on the convention of play. The usual rule considers that it is a draw, but Berlekamp considers that it is a loss, because the first player has usually an advantage.

It is also possible (like does David Wilson for example) to describe the ideal score s1/s2 as the difference s1-s2. The two presentations are equivalent.

An article about our method of computations and files with the resulting database will be available soon.